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Partially coherent sources with circular coherence

M. Santarsiero, R. Martínez-Herrero, D. Maluenda, J. C. G. de Sande, G. Piquero, F. Gori

2017
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Optics Letters
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A new class of partially coherent light sources is introduced. At the source plane, they exhibit perfect coherence along any annulus that is concentric to the source center. Between two points at different distances from the center, coherence can be partial or even vanishing. Such sources can be synthesized by using a generalized form of van Cittert-Zernike theorem where axial sources are used. Beam radiated by this type of sources are analyzed at the source plane and upon free propagation for
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... ome simple cases. OCIS codes: (030.1640) Coherence; (030.6600) Statistical optics. http://dx.doi.org/10.1364/ao.XX.XXXXXX Modeling partially coherent sources with different characteristics across the source plane and on propagation has been a big deal during the last decades . Experimental procedures aimed at synthesizing sources with tunable coherence properties have been also presented [22][23][24][25][26][27]. For a detailed review on the structure and synthesis of partially coherent sources, see [28] and references therein. Among the conceivable partially coherent sources, those of the Schell-model type [29] have played a crucial role in coherence theory. Such sources are characterized by a shift-invariant spectral degree of coherence [29] and then provide a useful model to represent many natural sources. Schell-model sources can be synthesized in a rather simple way, because they can be produced starting from a primary incoherent two-dimensional source and exploiting the van Cittert-Zernike theorem [29]. In this letter we present a class of planar, circularly symmetric, partially coherent sources, for which the degree of coherence between two points depends only on their distances from the source center. This means, in particular, that source regions containing points with high correlation have the shape of a donut and, in particular, the fields at any pair of points lying along a ring concentric with the source are mutually perfectly coherent. Natural sources endowed with such a property are rather unusual but, as will be shown, they can be synthesized in a very simple way. On the other hand, their propagation properties can be of interest, because their symmetry exactly reflects that of most optical systems used in practical applications. The proposed class of sources is characterized by a crossspectral density (CSD) at the source plane of the type where r is the modulus of the position vector on the source plane (r), while g and τ are complex-valued functions. The parameter δ µ has been introduced as a positive quantity having dimensions of a length, in order for g to be a function of a dimensionless argument. As we shall see, δ µ can be related to the coherence properties of the source. Note that CSDs written as in Eq. (1) fulfill the non-negativeness condition, and therefore represents physically realizable sources, for any g having non-negative Fourier transform [29]. Here and in the following we shall omit the explicit dependence from the temporal frequency. The irradiance profile of the source, namely, I 0 (r) = W 0 (r, r), is completely determined by the function τ(r), while the degree of spectral coherence between the points r 1 and r 2 , defined as [29] µ 0 (r 1 , r 2 ) = W 0 (r 1 , r 2 ) W 0 (r 1 , r 1 ) W 0 (r 2 , r 2 ) , is related to the function g. More precisely, it turns out that where, without loss of generality, we set g(0) = 1. In particular, the modulus of the degree of coherence depends only on the difference between the squared radial coordinates of the two points, and is one for any pairs of points lying on a circle centered on the axis origin. The analytical form of such a CSD reminds that of a source of the Schell-model type, but in the present case the dependence of the modulus of the degree of coherence is through the difference r 2 2 − r 2 1 , instead that through the difference of the position vectors themselves (r 2 − r 1 ). To illustrate some general coherence properties of this kind of sources, we first consider the case where one of the two points coincides with the axis origin. Then, the parameter δ µ can be defined in such a way to represent the maximum distance from the origin for which a significant correlation exists between the field values at the two points. In this way, the quantity πδ 2 µ provides an estimate of the coherence area of the source. In a

doi:10.1364/ol.42.001512
pmid:28409785
fatcat:6lydea46f5b53mw372v2ebmnq4